Chapter 2
Linear Equations
One of the problems encountered most frequently in scientific computation is the
solution of systems of simultaneous linear equations. This chapter covers the solu-
tion of linear systems by Gaussian elimination and the sensitivity of the solution to
errors in the data and roundoff errors in the computation.
2.1 Solving Linear Systems
With matrix notation, a system of simultaneous linear equations is written
Ax = b.
In the most frequent case, when there are as many equations as unknowns, A is a
given square matrix of order n, b is a given column vector of n components, and x
is an unknown column vector of n components.
Students of linear algebra learn that the solution to Ax = b can be written
x = A
−1
b, where A
−1
is the inverse of A. However, in the vast majority of practical
computational problems, it is unnecessary and inadvisable to actually compute
A
−1
. As an extreme but illustrative example, consider a system consisting of just
one equation, such as
7x = 21.
The best way to solve such a system is by division:
x =
21
7
= 3.
Use of the matrix inverse would lead to
x = 7
−1
× 21 = 0.142857 × 21 = 2.99997.
The inverse requires more arithmetic—a division and a multiplication instead of
just a division—and produces a less accurate answer. Similar considerations apply
December 26, 2005
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